Ranking Functions for Vector Addition Systems

Vector addition systems are an important model in theoretical computer science and have been used for the analysis of systems in a variety of areas. Termination is a crucial property of vector addition systems and has received considerable interest in the literature. In this paper we give a complete method for the construction of ranking functions for vector addition systems with states. The interest in ranking functions is motivated by the fact that ranking functions provide valuable additional information in case of termination: They provide an explanation for the progress of the vector addition system, which can be reported to the user of a verification tool, and can be used as certificates for termination. Moreover, we show how ranking functions can be used for the computational complexity analysis of vector addition systems (here complexity refers to the number of steps the vector addition system under analysis can take in terms of the given initial vector)

Cross-sectional associations between movement behaviors and cardio-metabolic risk markers in Danish children.

<p>Abbreviations: PA, physical activity; MVPA, moderate-to-vigorous physical activity; CSHQ, Children’s Sleep Habits Questionnaire; WC, waist circumference; MAP, mean arterial blood pressure; HOMA_IR, homeostatic model assessment of central insulin resistance; HDL-C, high density lipoprotein cholesterol; MetS-score, metabolic syndrome score.</p><p>Data are presented as unstandardized regression coefficients (β) with 95% confidence intervals (CI) using a linear mixed model with school and subject as random effects. The five cardio-metabolic risk markers were adjusted for baseline age, sex, pubertal status, and sex-pubertal status interaction (<b>Model 1</b>).</p><p>Coefficients represent the change in the outcome for a 100-cpm change in total PA, a 10-minute change in time spent in MVPA, a 1% change in sedentary time, a 60-minute change in screen time and sleep duration and a 1-point change in CSHQ. *, P<0.05; **, P<0.001.</p>¥<p>P<0.05 in <b>Model 2</b>: Model 1+ mutual adjustments between MVPA, sedentary time and sleep duration. Total PA was only adjusted for sleep duration.</p>£<p>P<0.05 in <b>Model 3</b>: Model 2+ fat mass index.</p>1<p>MAP was also adjusted for height.</p>2<p>HOMA_IR and triglycerides were log transformed.</p>3<p>MetS-score = (z-scores by baseline age, sex, pubertal status and sex-pubertal status interaction of) WC + MAP + HOMA_IR + triglycerides – HDL-C.</p

Eicosapentaenoic Acid and Docosahexaenoic Acid in Whole Blood Are Differentially and Sex-Specifically Associated with Cardiometabolic Risk Markers in 8–11-Year-Old Danish Children

<div><p>n-3 long-chain polyunsaturated fatty acids improve cardiovascular risk markers in adults. These effects may differ between eicosapentaenoic acid (EPA, 20∶5n-3) and docosahexaenoic acid (DHA, 22∶6n-3), but we lack evidence in children. Using baseline data from the OPUS School Meal Study we 1) investigated associations between EPA and DHA in whole blood and early cardiometabolic risk markers in 713 children aged 8–11 years and 2) explored potential mediation through waist circumference and physical activity and potential dietary confounding. We collected data on parental education, pubertal stage, 7-day dietary records, physical activity by accelerometry and measured anthropometry, blood pressure, and heart rate. Blood samples were analyzed for whole blood fatty acid composition, cholesterols, triacylglycerol, insulin resistance by the homeostatic model of assessment (HOMA-IR), and inflammatory markers. Whole blood EPA was associated with a 2.7 mmHg (95% CI 0.4; 5.1) higher diastolic blood pressure per weight% EPA, but only in boys. Heart rate was negatively associated with both EPA and DHA status (P = 0.02 and P = 0.002, respectively). Whole blood EPA was negatively associated with triacylglycerol (P = 0.003) and positively with total cholesterol, low density and high density lipoprotein (HDL) cholesterol and HDL:triacylglycerol (all P<0.01) whereas DHA was negatively associated with insulin and HOMA-IR (P = 0.003) and tended to be negatively associated with a metabolic syndrome-score (P = 0.05). Adjustment for waist circumference and physical activity did not change the associations. The association between DHA and HOMA-IR was attenuated but remained after adjustment for fiber intake and none of the other associations were confounded by dietary fat, protein, fiber or energy intake. This study showed that EPA status was negatively associated with triacylglycerol and positively with cholesterols whereas DHA was negatively associated with insulin resistance, and both were inversely associated with heart rate in children. The sex-specific associations with blood pressure confirm our previous findings and warrant further investigation.</p></div

Cardiometabolic risk markers in the children.

<p>Values are mean±SD or median (25^th–75^th percentile). HDL, high density lipoprotein; HOMA-IR, homeostatic model assessment-insulin resistance; LDL, low density lipoprotein.</p><p>^*Different from girls, *<i>P</i><0.05, **<i>P</i>>0.01, ***<i>P</i><0.001.</p>1<p><i>n</i> = 338 girls and <i>n</i> = 367 boys.</p>2<p><i>n</i> = 342 girls and <i>n</i> = 368 boys.</p><p>Cardiometabolic risk markers in the children.</p

Whole blood eicosapentaenoic acid is sex-specifically associated with diastolic blood pressure whereas docosahexaenoic acid is negatively associated with HOMA-IR in all children.

<p>Regression lines and 95% CI are shown for the associations between eicosapentaenoic acid and diastolic blood pressure in boys, <i>β</i> = 2.9, <i>P</i> = 0.02, <i>n</i> = 366 (A) and girls, <i>β</i> = −1.6, <i>P</i> = 0.28, <i>n</i> = 340 (B) and for the association between docosahexaenoic acid and HOMA-IR in the total study population, <i>β</i> = −0.03, <i>P</i> = 0.002, <i>n</i> = 713 (C). Plots were adjusted for school, class, parental education, age, height, puberty, and total fatty acids in whole blood. Blood pressure plots were additionally adjusted for ambient temperature and blood pressure device.</p

Fatty acid composition of children’s whole blood samples.

<p>Values are mean±SD or median (25^th–75^th percentile). Different from girls, *<i>P</i><0.05, ***<i>P</i><0.001.</p>1<p><i>n</i> = 342 girls for EPA, due to one extreme outlier.</p><p>Fatty acid composition of children’s whole blood samples.</p

Sociodemographic, anthropometric, and lifestyle characteristics of the children.

<p>Values are mean±SD or median (25^th–75^th percentile) unless stated otherwise. BMI, body mass index; LCPUFA, long-chain polyunsaturated fatty acids; NW, normal weight; OB, obese; OW, overweight; UW, underweight.</p><p>Different from girls, *<i>P</i><0.05, **<i>P</i><0.01, ***<i>P</i><0.001.</p>1<p>Corresponding to self-reported Tanner stage 2 or higher <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0109368#pone.0109368-Morris1" target="_blank">[19]</a>.</p>2<p><i>n</i> = 342 girls and <i>n</i> = 370 boys.</p>3<p>Based on age- and sex-specific cut-offs defined by Cole et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0109368#pone.0109368-Cole1" target="_blank">[27]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0109368#pone.0109368-Cole2" target="_blank">[28]</a>.</p>4<p><i>n</i> = 342 girls and <i>n</i> = 368 boys.</p>5<p><i>n</i> = 334 girls and <i>n</i> = 356 boys.</p>6<p><i>n</i> = 301 girls and <i>n</i> = 331 boys.</p><p>Sociodemographic, anthropometric, and lifestyle characteristics of the children.</p